Final answer:
The force constant between x₁ = 39 cm and x₂ = 67 cm results in -592.14 N/m, which does not match any of the provided options.
Step-by-step explanation:
To find the force constant between x₁ = 39 cm and x₂ = 67 cm, we must recognize that the spring constant k is essentially the slope of the force versus extension curve for the spring. Since we are given different spring constants for different extensions of the same spring, it implies that the spring does not follow Hooke's law over the entire range of its extension, and we are dealing with a variable stiffness spring.
The force exerted by the spring at x₁ = 39 cm can be calculated using the spring constant k₁ = 720 N/m:
F₁ = k₁ × x₁
F₁ = 720 N/m × 0.39 m
F₁ = 280.8 N
The force at x₂ = 67 cm is constant at F3 = 115 N, which is the force at x₃ = 91 cm. The spring force between x₁ and x₂ changes linearly, so we can use the two points (x₁, F₁) and (x₂, F₃) to find the spring constant k₂ for that range:
k₂ = (F₃ - F₁) / (x₂ - x₁)
k₂ = (115 N - 280.8 N) / (0.67 m - 0.39 m)
k₂ = (-165.8 N) / (0.28 m)
k₂ = -592.14 N/m
The negative sign indicates that the force exerted by the spring actually decreases as it stretches from x₁ to x₂. Nevertheless, the magnitude of this spring constant does not match any of the options provided (A) 78 N/m, (B) 120 N/m, (C) 80 N/m, (D) 105 N/m. Hence, there seems to be a mistake in the question or the given options.